Some results on transcendental entire solutions of certain nonlinear differential-difference equations

TL;DR

This paper investigates transcendental entire solutions of certain nonlinear differential-difference equations, providing new results that improve and extend previous findings in the field.

Contribution

The paper offers new theorems characterizing solutions of specific nonlinear differential-difference equations, advancing understanding of their transcendental entire solutions.

Findings

Identified conditions for transcendental entire solutions to exist.

Extended previous results to more general equations.

Provided explicit forms and properties of solutions.

Abstract

In this paper, we study the transcendental entire solutions for the nonlinear differential-difference equations of the forms: $f^{2}(z)+\widetilde{\omega} f(z)f'(z)+q(z)e^{Q(z)}f(z+c)=u(z)e^{v(z)}$, and $f^{n}(z)+\omega f^{n-1}(z)f'(z)+q(z)e^{Q(z)}f(z+c)=p_{1}e^{\lambda_{1} z}+p_{2}e^{\lambda_{2} z}, \quad n\geq 3,$ where $\omega$ is a constant, $\widetilde{\omega}, c, \lambda_{1}, \lambda_{2}, p_{1}, p_{2}$ are non-zero constants, $q, Q, u, v$ are polynomials such that $Q,v$ are not constants and $q,u\not\equiv0$. Our results are improvements and complements of some previous results.